Date: Fri, 11 Nov 1994 13:57:32 -0800 From: Y V Srinivas <srinivas@kestrel.edu>
Do (finite) colimits exist in categories of diagrams?
Here are the definitions of the terms used above.
A diagram of shape S in a category C is a functor D: S -> C.
A covariant diagram morphism from D1: S1 -> C to D2: S2 -> C is a pair <sm, mu> consisting a shape morphism (functor) sm: S1 -> S2 and a natural transformation mu: D1 -> D2 o sm.
A contravariant diagram morphism is similar except that the direction of the natural transformation is opposite.
We thus get two categories of diagrams in C (of all shapes). Mac Lane calls these "super comma categories" (p.111).
Now, the question reads: given that (finite) colimits exist in C, do (finite) colimits exist in the two categories of diagrams in C?
I haven't been able to come up with a construction or a proof that colimits don't exist. I appreciate any help you can provide on this matter. Also, I would like to know if there are related results, e.g., with limits.
- srinivas In my thesis, Dalhousie 1976, I used a (2-)category of diagrams of the kind that you describe, with C=set. The 2-functor cat--->CAT which sends S to set^(S^op) gives rise to a fibration, say P:D_--->cat, and the objects of D_ are evidently diagrams in set of variable shape. If you regard such an object, D:S^op--->set, as a discrete fibration, say D'--->S, then it is straightforward to describe a 2-fully faithful 2-functor, I:D_--->cat^2 . Now codomain:cat^2--->cat is itself a fibration over cat and the inclusion I is a 2-functor over cat. I has a left adjoint, L, that preserves finite products, from which it follows that D_ is complete, cocomplete and cartesian closed, as a 2-category. By examining the nature of these constructions for my D_ you should be able to find what you need of a general C to answer your question.
Well, to continue somewhat, P above clearly preserves limits but it also preserves colimits because the adjunction L-|I is an adjunction over cat. This should help to get started on the general problem. In fact, P preserves exponentiation too. (Pseudo-)category objects in D_ are interesting. Necessarily, they give rise to pseudo-category objects in cat. If, for example, V is a monoidal category then P-over the "category object" VxV--->V<---1 in cat are precisely the set^(V^op)-categories, where the monoidal structure of set^(V^op) is Brian Day's (universal) convolution structure. But there are many other "category objects" in cat that can be constructed from V and its monoidal data. Assembling these and looking at their inverse image in D_ allows one to construct a variant of the Pare-Schumacher theory of indexed categories that is useful when a given monoidal structure on the base for indexing is needed for multiply indexed families. RJ Wood